Equation Calculator with Steps
Solve one-variable linear and quadratic equations. Enter x on either side, use fractions or decimals, and review the standard form, exact solutions, decimal values and substitution checks.
Last Updated: July 26, 2026
Online Equation Solver
Enter one equation using x. The calculator simplifies both sides, moves all terms to the left, identifies the degree and solves a constant, linear or quadratic result.
Your equation solution will appear here.
Calculation steps
- Enter a one-variable equation with one equals sign.
Solution details
| Solution | Exact form | Decimal form | Check |
|---|---|---|---|
| Solve an equation to see each solution. | |||
Recent calculations
- Your last six results will appear here.
How to Use This Equation Calculator
- Enter an equation with one equals sign and use x as the variable.
- Keep Auto-detect selected for normal use. Choose Linear or Quadratic only when you want to enforce a specific equation type.
- Select the number of decimal places for approximate answers.
- Select Solve Equation. Review the standard form, exact solution set, decimals, discriminant and steps.
- Check the solution table. Linear and rational quadratic roots are substituted exactly. Irrational and complex roots receive an algebraic residual check.
You may enter integers, finite decimals, fractions, scientific notation, parentheses and whole-number powers from 1 to 12 for variable expressions. Constant bases may use exponent 0, except that 0^0 is undefined. The parser recognizes implicit multiplication, so 3x and 3*x mean the same thing. Write 2500 instead of 2,500 because commas are not accepted inside numbers.
What This Equation Solver Does
The calculator reads each side as a one-variable polynomial expression. It expands supported products, combines like terms and subtracts the right side from the left. This produces a standard equation P(x) = 0. The degree of P determines the solution method.
A constant result becomes an identity or contradiction. A first-degree result uses linear isolation. A second-degree result uses the quadratic formula. Equations that look complicated may still qualify after simplification. For example, x(x + 2) = x^2 + 10 reduces to 2x - 10 = 0, so it is linear.
This page handles one equation in one variable. A future System of Equations Calculator will handle two or more equations together. A dedicated Quadratic Equation Calculator will provide coefficient entry and deeper quadratic analysis. This general solver focuses on typed equations, automatic classification and clear steps.
Equation vs Expression
An expression represents a value, while an equation states that two expressions are equal. The expression 3x + 2 has no equals sign and does not ask for a solution by itself. The equation 3x + 2 = 14 asks which value of x makes the two sides equal.
This distinction controls the tool. Enter a complete left side, one equals sign and a complete right side. If your task asks you to expand, combine like terms or evaluate an expression without solving an equality, use the separate Algebra Calculator after that page is published.
Supported Equation Types
| Equation type | Example | Result |
|---|---|---|
| Linear, one side | 3x + 2 = 14 | Solves x = 4 and verifies both sides equal 14. |
| Linear, both sides | 5x - 3 = 2x + 9 | Moves terms, solves x = 4 and checks the original equation. |
| Linear with fractions | (1/2)x + 3 = 7 | Keeps the coefficient exact and returns x = 8. |
| Quadratic with two real roots | x^2 - 5x + 6 = 0 | Returns x = 2 and x = 3. |
| Quadratic with one repeated root | x^2 - 6x + 9 = 0 | Returns x = 3 as one distinct repeated root. |
| Quadratic with complex roots | x^2 + 4 = 0 | Returns x = -2i and x = 2i. |
| Identity | 2(x + 1) = 2x + 2 | Reports infinitely many real solutions. |
| Contradiction | 2x + 1 = 2x + 3 | Reports no solution. |
Linear Equation Formula
After simplification, a linear equation has the form ax + b = 0, where a is not zero. Isolate the variable term, then divide by its coefficient.
ax = -b
x = -b/a
For 3(x - 2) + 4 = 2x + 9, the left side becomes 3x - 2. Subtracting the right side gives x - 11 = 0. Therefore x = 11. Substitution gives 31 on both original sides.
Properties of Equality
Solving an equation preserves equality by applying the same valid operation to both sides. You may add or subtract the same number, multiply both sides by the same number, or divide both sides by the same nonzero number. These actions produce an equivalent equation with the same solution set.
For 5x - 3 = 2x + 9, subtract 2x from both sides to get 3x - 3 = 9. Add 3 to both sides to get 3x = 12. Divide both sides by 3 to obtain x = 4. The calculator may display the equivalent standard form 3x - 12 = 0, then isolate x from that form.
Multiplying by zero or dividing by an expression that might be zero can change a solution set. This is why the parser rejects variable denominators instead of removing them without tracking excluded values.
Quadratic Equation Formula
A quadratic equation has the standard form ax^2 + bx + c = 0 with a not equal to zero. Its discriminant is D = b^2 - 4ac. The sign of D controls the root pattern.
x = (-b ± √D) / (2a)
- D > 0 gives two distinct real roots.
- D = 0 gives one distinct repeated real root.
- D < 0 gives two complex conjugate roots.
Fraction coefficients are cleared by multiplying every coefficient by a common denominator. This keeps the same solution set and makes the discriminant an integer. The displayed standard form still preserves the equation represented by your input.
Worked Example: Two Real Roots
Solve x^2 - 5x + 6 = 0. Here a = 1, b = -5 and c = 6. The discriminant is 25 - 24 = 1.
x = (5 + 1) / 2 or (5 - 1) / 2
x = 3 or x = 2
Both answers verify exactly. Substituting 2 produces 4 - 10 + 6 = 0. Substituting 3 produces 9 - 15 + 6 = 0.
Worked Example: Irrational Roots
For x^2 - 2 = 0, the discriminant is 8. The radical simplifies because 8 = 4 times 2. The two exact roots are -sqrt(2) and sqrt(2). Their decimal values are about -1.414213562373 and 1.414213562373 when 12 places are selected.
The approximation mark matters. A rounded decimal is useful for measurement or plotting, but it is not the exact algebraic solution. Keep the simplified radical when the problem asks for an exact answer.
Identities and Contradictions
If every term cancels, the equation becomes 0 = 0. This is an identity, so every real x satisfies it. If all variable terms cancel but a nonzero constant remains, the equation becomes a false statement such as -2 = 0. That contradiction has no solution.
The calculator labels these cases before attempting division. This prevents a divide-by-zero error from being mistaken for an answer.
Exact Fractions, Radicals and Decimals
Finite decimals and written fractions are stored as reduced rational numbers backed by integers. For example, 0.125 becomes 1/8. Linear answers and rational quadratic roots remain exact fractions.
When a quadratic root is irrational, the exact result uses a square-root form and simplifies common square factors when detected. When the discriminant is negative, the result uses i, where i^2 = -1. Decimal approximations are calculated separately and never replace the exact form.
Input Rules and Calculator Limits
Use x as the only variable. The expression parser supports +, -, *, /, ^, parentheses, implicit multiplication, integers, finite decimals, fractions and scientific notation. Division is allowed only by a nonzero constant expression. This means (x + 2)/2 is supported, but 1/x is not.
The calculator expands input safely to inspect the final degree, but it solves only results with degree 0, 1 or 2. Cubic and higher-degree equations, variable denominators, negative exponents, radical variables, absolute values, logarithms and trigonometric functions are outside this solver. Unsupported input receives a specific message and stays in the field for correction.
An equation may contain up to 300 characters and 240 parsed tokens. Each written number may contain up to 50 significant digits. Scientific exponents must stay from -100 through 100. Entered powers and intermediate polynomial degree may not exceed 12. Deeply nested or oversized exact calculations stop with an error instead of returning a partial result.
Common Equation-Solving Mistakes
- Changing the sign of a term on only one side instead of performing the same operation on both sides.
- Failing to distribute a factor to every term inside parentheses.
- Combining x and x^2 as if they were like terms.
- Using the linear formula before checking whether a quadratic term remains.
- Forgetting the plus-or-minus sign in the quadratic formula.
- Calculating b^2 - 4ac with the wrong sign when b or c is negative.
- Rounding an intermediate fraction or radical before the final step.
- Accepting an answer without substituting it into the original equation.
Method and Review Basis
The simplification and equation rules follow the algebra essentials in OpenStax College Algebra 2e. The linear process follows the general strategy in OpenStax Intermediate Algebra 2e. Quadratic classification and the formula follow OpenStax Quadratic Equations.
The page parses a defined grammar instead of executing the typed text as JavaScript. Calculations run in the current browser page. Equation text and recent history are not submitted to a calculation server, and the history disappears when the page reloads.
Related Calculators
Frequently Asked Questions
What equations does this calculator solve?
It solves one-variable equations that simplify to a constant, linear or quadratic polynomial equation in x.
Can x appear on both sides?
Yes. The calculator simplifies both sides, subtracts the right side from the left and solves the resulting standard equation.
Does it solve quadratic equations?
Yes. It applies the quadratic formula when the simplified equation has degree 2 and returns real, repeated or complex roots.
Can I enter fractions and decimals?
Yes. Enter fractions such as 1/2, finite decimals such as 0.25, or scientific notation such as 2e3. Rational arithmetic stays exact.
What does infinitely many solutions mean?
Both sides simplify to the same expression, so every real value of x satisfies the equation.
What does no solution mean?
The variable terms cancel but unequal constants remain, producing a contradiction that no value of x satisfies.
What is the discriminant?
For ax^2 + bx + c = 0, the discriminant is b^2 - 4ac. Its sign determines whether the quadratic has two real roots, one repeated root or two complex roots.
How are irrational roots displayed?
The calculator shows an exact square-root form, simplifies common square factors when detected, and provides a separate decimal approximation.
Does it show complex solutions?
Yes. A negative quadratic discriminant produces two conjugate roots written with i and separate decimal approximations.
Why is division by x rejected?
Division by x creates a rational equation with domain restrictions. This calculator supports division only by a nonzero constant expression.
Why does it reject a cubic equation?
The solver is intentionally limited to equations that simplify to degree 2 or lower. It does not guess higher-degree roots.
Is my equation sent to a server?
No input is needed by a server to calculate the answer. Parsing, solving and recent history run in the current browser page.
Disclaimer: This calculator provides general mathematical results for learning and checking work. Verify notation, domain restrictions, assumptions and required rounding before using an answer for academic, scientific, financial, engineering or other important decisions.